Dynamical Systems in Neuroscience:

Electrophysiology of Neurons 29If there are no additional current sources or sinks, such as synaptic current, axialcurrent, tangential current along the membrane surface, or current injected via anelectrode, then I = 0. In this case, the membrane potential is typically bounded bythe equilibrium potentials in the following order (see Fig. 2.4):E K < E Cl < V (at rest)< E Na < E Ca ,so that I Na , I Ca < 0 (inward currents) and I K , I Cl > 0 (outward currents). From(2.2) it follows that inward currents increase the membrane potential, i.e., make itmore positive (depolarization), whereas outward currents decrease it, i.e., make it morenegative (hyperpolarization). Notice that I Cl is called an outward current even thoughthe flow of Cl − ions is inward; the ions bring negative charge inside the membrane,which is equivalent to positively charged ions leaving the cell, as in I K .2.1.4 Resting potential and input resistanceIf there were only K + channels, as in Fig. 2.2, the membrane potential would quicklyapproach the K + equilibrium potential, E K , which is around −90 mV. Indeed,C ˙V = −I K = −g K (V − E K )in this case. However, most membranes contain a diversity of channels. For example,Na + channels would produce an inward current and pull the membrane potential towardsthe Na + equilibrium potential, E Na , which could be as large as +90 mV. Thevalue of the membrane potential at which all inward and outward currents balance eachother so that the net membrane current is zero corresponds to the resting membranepotential. It can be found from the equation (2.3, I = 0) by setting ˙V = 0. Theresulting expressionV rest = g NaE Na + g Ca E Ca + g K E K + g Cl E Clg Na + g Ca + g K + g Cl(2.4)has a nice mechanistic interpretation: V rest is the center of mass of the balance depictedin Fig. 2.4. Incidentally, the entire equation (2.3) can be written in the formwhereC ˙V = I − g inp (V − V rest ) , (2.5)g inp = g Na + g Ca + g K + g Clis the total membrane conductance, called input conductance. The quantity R inp =1/g inp is the input resistance of the membrane, and it measures the asymptotic sensitivityof the membrane potential to injected or intrinsic currents. Indeed, from (2.5) itfollows thatV → V rest + IR inp , (2.6)